Many teachers ask students what we call
"True/False" in worksheets and on tests and quizzes. A "true or false
question" might look like this: " The diagonals of a parallelogram
bisect each other." The correct answer is "True". The student has a
50 per cent chance of guessing and coming up with the right answer.

A question about the properties of a parallelogram
could, instead, be asked in this way: "You are given the following
statement, and are to decide if the statement is always true,
sometimes true, or never true: "The diagonals of a rhombus are
congruent." The answer would be "The diagonals of a
parallelogram are sometimes congruent", or simply "Sometimes".
Phrasing the question in this way gives the students a one in three
chance of guessing the answer, and therefore is a better test of his
knowlege.

But a more interesting question would be as
follows: "Given the sentence "The diagonals of a parallelogram are
congruent" decide if the sentence is always true (true for any
parallelogram), sometimes true, or never true, and explain why." This
not only eliminates guessing, it causes the student to think
carefully about not only what the answer is, but why.
It also gives the student practice in an informal type of proof. Of
course, this all depends on the definition that your text book uses;
in this case we are defining a parallelogram as a polygon with
exactly two pair of parallel sides.

A student might answer this question as follows:
The definition of a paralellogram is a quadrilateral with both
pairs of opposite sides parallel. But this does not mean that one
pair must be congruent to the other pair, as you can see in the
diagram below:

If the parallelogram happens to have all angles
right angles, which makes it a rectangle, and in that case the
diagonals are congruent as shown:

Therefore the answer is "SOMETIMES".
Another example of an "ASN Explain" question might be the
following:

If all sides of a polygon are congruent than
all of the interior angles are congruent. Is this statement always,
sometimes, or never true?

A student might answer as
follows:

"If all sides of a polygon are congruent, it is
only sometimes true that all of the interior
angles are congruent. (If a triangle is equilateral then it is
equiangular.) But, let's look at a hexagon for example, as in the
drawing below: All of its sides are congruent, and all the interior
angles appear to be congruent also.

But in the hexagon below, all the sides are
congruent, but that the angles are clearly not all congruent to each
other:"

The student might even choose to take this
explanation further: "This is not only true for hexagons, but even
for quadrilaterals every other polygon with the single exception of
the triangle." (If a triangle is equilateral then it is
equiangular.)

Asking the question in this way gives students
opportunities to make conjectures and then test them, and to explore
the issue by sketching various figures, making judgments and
conclusions on his or her own rather than referring to a memorized
list of rules.

In their reflections on this process, students
expressed their thoughts about doing projects that were "outside the
box", and not just problems directly from the textbook. Lina said
this about the problem above: "When you read a theorem in the book
you just believe it, because the book is always right. And it's easy
to think that if a theorem works for one figure it will be true for
another. This assignment really made me think again about that. I
guess we have to keep our brains working, and not just make
assumptions. What is true for a triangle is not necessarily true for
anything else!"

And Jordan had this to say: "When you think
about it, there are so many ways to look at math. And you really have
to pay attention because what happens in one situation isn't what
will happen in another. Hmmm. That's pretty true with life in
general, isn't it!"

A somewhat related exercise is the one we did on "Proving
Quadrilaterals Congruent". The idea again was to compare what is true
about triangles with what is true about other geometric figures. As
with the hexagons in the project above, the information one needs to
be sure two quadrilaterals (or any other pair of polygons) congruent
is different for each type of polygon.

The central question was this: "What is it
about triangles that allows us to use these 3-letter "shortcuts" to
prove them congruent? Why do we NOT have to prove all 6 pairs of
sides and angles? Could you use these same shortcuts to prove two
quadrilaterals congruent? If not, why not? What would it take to
prove two quadrilaterals congruent? Always, If 3 sides of one
quadrilateral is congruent to 3 sides of another quadrilateral are
the quadrilaterals Always, Sometimes or Never congruent"?
The students worked in groups of three or four to experiment and
come to some conclusions.

Adam, Leah, Sara and Stacy began with these
thoughts and conclusions: "In a triangle, there are a total of 6
sides and angles. In the three letter SAS, ASA etc. "shortcuts", you
only have to prove 3 pairs of parts congruent to prove the triangles
congruent. So you really only need to be given 3 pieces of
information, like 3 sides, or 2 sides and the angle between. Once you
have half (3 out of 6) pieces of information, you can prove the
triangles congruent. You can think of a quadrilateral as being made
up of 2 triangles. Therefore it would seem that you would need six
pieces of information (sides and/or angles) to prove a pair of quads
congruent.

Actually, we found out that you only need 5 pieces
of information (sides and/or angles). The reason for this is that all
quadrilaterals are made up of 2 triangles, if you just draw any
diagonal."

In explaining how they did this, one group (Adam,
Leah, Sara and Stacie) said: "If you can prove the two triangles
of the one quadrilaterals congruent to the two triangles of the other
quadrilateral, then you will have proved the two quadrilaterals
congruent. Since it only takes 3 pieces of information (2 sides and
an angle, for example) to prove a pair of triangles congruent, and
since in each quadrilateral there a pair of triangles, there are 6
pieces of information except that the diagonal is shared, so there
are only 5 pieces of information needed to prove the 2 quadrilaterals
congruent. And, if you said "SAS" for triangles, we could use the
same type of lettering for the quads and say: that the following
5-letter "shortcuts" would do it: SASAA, SASSS, SASAS, and
ASASA".

The students proved each of these separate cases.
You can see the proof of SASSS below:

In her reflection on this project, Nella said
"This assignment really pushed my mind to its fullest. We had to
try to decide if it was even possible to prove something, and if so,
how? This challenged us beyond what we were taught; we really were
"out there" in a new world!"

"No human
investigation can be called real science if it cannot be demonstrated
mathematically."